Question

For a positively skewed binomial distribution with n = 20, which of the following values might be the value of mean?
A. $8$
B. $10$
C. $15$
D. $20$

Answer 1

Hint: According to given in the question we have to determine the value of mean for a positively skewed binomial distribution with n = 20. So, first of all we have to understand about the positively skewed binomial distribution which is as explained below:
According to positively skewed binomial distribution a distribution is skewed if one of its tails is longer as compared to the other.
Now, we have to determine the probability of success which can be determined by dividing the required event by the total number of possible outcomes or the sample space.
Now, we have to consider this as a positive skewed binomial distribution as mentioned in the question.
As mentioned for a positively skewed binomial distribution with n = 20, we have to check it for the probability and then we have to compare them.

Complete step-by-step solution:
Step 1: First of all we have to determine the probability of success which can be determined by dividing the required event by the total number of possible outcomes or the sample space as mentioned in the solution hint, so here our required event is 1 and the total number of possible outcomes are 2. Hence,
Probability$ = \dfrac{1}{2}$
Which can be considered as $P < 0.5$ which means probability is less then 0.5 and P represent the probability.
Step 2: Now, as mentioned for a positively skewed binomial distribution with n = 20, we have to check it for the probability and then we have to compare them. Hence,
$ \Rightarrow P < 0.5$ and,
$ \Rightarrow n = 20$
Step 3: Now, in binomial we have to determine the mean as:
$
   \Rightarrow {n_P} < 20 \times (0.5) \\
   \Rightarrow {n_P} < 10
 $
Step 4: Now, to obtain the required mean with the help of ${n_P} < 10$ which is as obtained in the solution step 3 we have to take the value which should be less than 10 and as mentioned in the options 8 is less than 10. Hence,
The value of mean = 8
Hence, we have to determine the value of mean = 8 for a positively skewed binomial distribution with n = 20.

Therefore option (A) is correct.

Note: Distribution with positive skews are more common than the distribution of negative skews and the mean that it has a long tail in the positive.
Binomial distribution is skewed if one of its tails is longer as compared to the other.